TSTP Solution File: SWW225+1 by Twee---2.5.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.5.0
% Problem : SWW225+1 : TPTP v8.2.0. Released v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee /export/starexec/sandbox/benchmark/theBenchmark.p --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon Jun 24 18:26:36 EDT 2024
% Result : Theorem 169.50s 21.77s
% Output : Proof 170.37s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SWW225+1 : TPTP v8.2.0. Released v5.2.0.
% 0.07/0.13 % Command : parallel-twee /export/starexec/sandbox/benchmark/theBenchmark.p --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding
% 0.14/0.35 % Computer : n009.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Jun 19 05:49:09 EDT 2024
% 0.14/0.35 % CPUTime :
% 169.50/21.77 Command-line arguments: --flatten
% 169.50/21.77
% 169.50/21.77 % SZS status Theorem
% 169.50/21.77
% 169.50/21.78 % SZS output start Proof
% 169.50/21.78 Take the following subset of the input axioms:
% 170.37/21.79 fof(arity_Complex__Ocomplex__Fields_Ofield, axiom, class_Fields_Ofield(tc_Complex_Ocomplex)).
% 170.37/21.79 fof(arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra, axiom, class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex)).
% 170.37/21.79 fof(arity_RealDef__Oreal__Fields_Ofield, axiom, class_Fields_Ofield(tc_RealDef_Oreal)).
% 170.37/21.79 fof(arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra, axiom, class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal)).
% 170.37/21.79 fof(conj_0, conjecture, c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, hAPP(c_RealDef_Oreal(tc_Nat_Onat), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), hAPP(c_RealDef_Oreal(tc_Nat_Onat), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))))).
% 170.37/21.79 fof(fact_Suc__le__mono, axiom, ![V_m_2, V_n_2]: (c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(V_n_2), c_Nat_OSuc(V_m_2)) <=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_n_2, V_m_2))).
% 170.37/21.79 fof(fact_fz_I1_J, axiom, c_SEQ_Osubseq(v_f____)).
% 170.37/21.79 fof(fact_real__of__nat__le__iff, axiom, ![V_n_2_2, V_m_2_2]: (c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, hAPP(c_RealDef_Oreal(tc_Nat_Onat), V_n_2_2), hAPP(c_RealDef_Oreal(tc_Nat_Onat), V_m_2_2)) <=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_n_2_2, V_m_2_2))).
% 170.37/21.79 fof(fact_seq__suble, axiom, ![V_fa_2, V_n_2_2]: (c_SEQ_Osubseq(V_fa_2) => c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_n_2_2, hAPP(V_fa_2, V_n_2_2)))).
% 170.37/21.79
% 170.37/21.79 Now clausify the problem and encode Horn clauses using encoding 3 of
% 170.37/21.79 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 170.37/21.79 We repeatedly replace C & s=t => u=v by the two clauses:
% 170.37/21.79 fresh(y, y, x1...xn) = u
% 170.37/21.79 C => fresh(s, t, x1...xn) = v
% 170.37/21.79 where fresh is a fresh function symbol and x1..xn are the free
% 170.37/21.79 variables of u and v.
% 170.37/21.79 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 170.37/21.79 input problem has no model of domain size 1).
% 170.37/21.79
% 170.37/21.79 The encoding turns the above axioms into the following unit equations and goals:
% 170.37/21.79
% 170.37/21.79 Axiom 1 (arity_Complex__Ocomplex__Fields_Ofield): class_Fields_Ofield(tc_Complex_Ocomplex) = true2.
% 170.37/21.79 Axiom 2 (arity_RealDef__Oreal__Fields_Ofield): class_Fields_Ofield(tc_RealDef_Oreal) = true2.
% 170.37/21.79 Axiom 3 (arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra): class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex) = true2.
% 170.37/21.79 Axiom 4 (arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra): class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal) = true2.
% 170.37/21.79 Axiom 5 (fact_fz_I1_J): c_SEQ_Osubseq(v_f____) = true2.
% 170.37/21.79 Axiom 6 (fact_Suc__le__mono): fresh1006(X, X, Y, Z) = true2.
% 170.37/21.79 Axiom 7 (fact_real__of__nat__le__iff): fresh286(X, X, Y, Z) = true2.
% 170.37/21.79 Axiom 8 (fact_seq__suble): fresh260(X, X, Y, Z) = true2.
% 170.37/21.79 Axiom 9 (fact_seq__suble): fresh260(c_SEQ_Osubseq(X), true2, Y, X) = c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, Y, hAPP(X, Y)).
% 170.37/21.79 Axiom 10 (fact_Suc__le__mono): fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, X, Y), true2, Y, X) = c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(X), c_Nat_OSuc(Y)).
% 170.37/21.79 Axiom 11 (fact_real__of__nat__le__iff): fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, X, Y), true2, Y, X) = c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, hAPP(c_RealDef_Oreal(tc_Nat_Onat), X), hAPP(c_RealDef_Oreal(tc_Nat_Onat), Y)).
% 170.37/21.79
% 170.37/21.79 Lemma 12: class_Fields_Ofield(tc_RealDef_Oreal) = class_Fields_Ofield(tc_Complex_Ocomplex).
% 170.37/21.79 Proof:
% 170.37/21.79 class_Fields_Ofield(tc_RealDef_Oreal)
% 170.37/21.79 = { by axiom 2 (arity_RealDef__Oreal__Fields_Ofield) }
% 170.37/21.79 true2
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 class_Fields_Ofield(tc_Complex_Ocomplex)
% 170.37/21.79
% 170.37/21.79 Lemma 13: class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex) = class_Fields_Ofield(tc_RealDef_Oreal).
% 170.37/21.79 Proof:
% 170.37/21.79 class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex)
% 170.37/21.79 = { by axiom 3 (arity_Complex__Ocomplex__RealVector_Oreal__normed__div__algebra) }
% 170.37/21.79 true2
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 class_Fields_Ofield(tc_Complex_Ocomplex)
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 class_Fields_Ofield(tc_RealDef_Oreal)
% 170.37/21.79
% 170.37/21.79 Lemma 14: class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal) = class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex).
% 170.37/21.79 Proof:
% 170.37/21.79 class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal)
% 170.37/21.79 = { by axiom 4 (arity_RealDef__Oreal__RealVector_Oreal__normed__div__algebra) }
% 170.37/21.79 true2
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 class_Fields_Ofield(tc_Complex_Ocomplex)
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 class_Fields_Ofield(tc_RealDef_Oreal)
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex)
% 170.37/21.79
% 170.37/21.79 Lemma 15: c_SEQ_Osubseq(v_f____) = class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal).
% 170.37/21.79 Proof:
% 170.37/21.79 c_SEQ_Osubseq(v_f____)
% 170.37/21.79 = { by axiom 5 (fact_fz_I1_J) }
% 170.37/21.79 true2
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 class_Fields_Ofield(tc_Complex_Ocomplex)
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 class_Fields_Ofield(tc_RealDef_Oreal)
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex)
% 170.37/21.79 = { by lemma 14 R->L }
% 170.37/21.79 class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal)
% 170.37/21.79
% 170.37/21.79 Goal 1 (conj_0): c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, hAPP(c_RealDef_Oreal(tc_Nat_Onat), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), hAPP(c_RealDef_Oreal(tc_Nat_Onat), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))))) = true2.
% 170.37/21.79 Proof:
% 170.37/21.79 c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, hAPP(c_RealDef_Oreal(tc_Nat_Onat), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), hAPP(c_RealDef_Oreal(tc_Nat_Onat), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))))
% 170.37/21.79 = { by axiom 11 (fact_real__of__nat__le__iff) R->L }
% 170.37/21.79 fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))), true2, c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))), class_Fields_Ofield(tc_Complex_Ocomplex), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))), class_Fields_Ofield(tc_RealDef_Oreal), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))), class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 14 R->L }
% 170.37/21.79 fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))), class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 15 R->L }
% 170.37/21.79 fresh286(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 10 (fact_Suc__le__mono) R->L }
% 170.37/21.79 fresh286(fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), true2, hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 fresh286(fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), class_Fields_Ofield(tc_Complex_Ocomplex), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 fresh286(fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), class_Fields_Ofield(tc_RealDef_Oreal), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 fresh286(fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 14 R->L }
% 170.37/21.79 fresh286(fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 15 R->L }
% 170.37/21.79 fresh286(fresh1006(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 9 (fact_seq__suble) R->L }
% 170.37/21.79 fresh286(fresh1006(fresh260(c_SEQ_Osubseq(v_f____), true2, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 fresh286(fresh1006(fresh260(c_SEQ_Osubseq(v_f____), class_Fields_Ofield(tc_Complex_Ocomplex), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 fresh286(fresh1006(fresh260(c_SEQ_Osubseq(v_f____), class_Fields_Ofield(tc_RealDef_Oreal), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 fresh286(fresh1006(fresh260(c_SEQ_Osubseq(v_f____), class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 14 R->L }
% 170.37/21.79 fresh286(fresh1006(fresh260(c_SEQ_Osubseq(v_f____), class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 15 R->L }
% 170.37/21.79 fresh286(fresh1006(fresh260(c_SEQ_Osubseq(v_f____), c_SEQ_Osubseq(v_f____), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____), v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 8 (fact_seq__suble) }
% 170.37/21.79 fresh286(fresh1006(true2, c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 fresh286(fresh1006(class_Fields_Ofield(tc_Complex_Ocomplex), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 fresh286(fresh1006(class_Fields_Ofield(tc_RealDef_Oreal), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 fresh286(fresh1006(class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 14 R->L }
% 170.37/21.79 fresh286(fresh1006(class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 15 R->L }
% 170.37/21.79 fresh286(fresh1006(c_SEQ_Osubseq(v_f____), c_SEQ_Osubseq(v_f____), hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 6 (fact_Suc__le__mono) }
% 170.37/21.79 fresh286(true2, c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by axiom 1 (arity_Complex__Ocomplex__Fields_Ofield) R->L }
% 170.37/21.79 fresh286(class_Fields_Ofield(tc_Complex_Ocomplex), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 12 R->L }
% 170.37/21.79 fresh286(class_Fields_Ofield(tc_RealDef_Oreal), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 13 R->L }
% 170.37/21.79 fresh286(class_RealVector_Oreal__normed__div__algebra(tc_Complex_Ocomplex), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 14 R->L }
% 170.37/21.79 fresh286(class_RealVector_Oreal__normed__div__algebra(tc_RealDef_Oreal), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.79 = { by lemma 15 R->L }
% 170.37/21.80 fresh286(c_SEQ_Osubseq(v_f____), c_SEQ_Osubseq(v_f____), c_Nat_OSuc(hAPP(v_f____, c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____))), c_Nat_OSuc(c_Groups_Oplus__class_Oplus(tc_Nat_Onat, v_N1____, v_N2____)))
% 170.37/21.80 = { by axiom 7 (fact_real__of__nat__le__iff) }
% 170.37/21.80 true2
% 170.37/21.80 % SZS output end Proof
% 170.37/21.80
% 170.37/21.80 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------